Extreme Preference and Distortion Risk Measures on Tail ......the extreme scenarios go to extremes....

Introduction and MotivationExtreme Preference

Distortion Risk Measure on Tail RegionsMain Result about Distortion Risk Measure

Extreme Preference and Distortion Risk Measureson Tail Regions with Nonparametric Inference

Method

Xing Wang

Department of MathematicsIllinois State University

Oct 5th

Xing WangExtreme Preference and Distortion Risk Measures on Tail Regions with Nonparametric Inference Method



Outline

1 Introduction and Motivation

2 Extreme PreferenceTail Risks under the Extreme ScenariosExtreme Preference and Its Distortion

3 Distortion Risk Measure on Tail RegionsDistortion Risk Measure DifinitionStatistical Inference

4 Main Result about Distortion Risk Measure




Learn risk exposure of a financial entity under the occurrenceof some other financial variables’ extreme scenarios.Example:

Measure the systemic risk of the expected loss on somefinancial equity return conditional on the occurrence of anextreme loss in the aggregated return of the financial sector;Measure the relative risk of individual financial entity to somebenchmark as a sensitive monitoring index of the marketco-movement.

Traditional risk measures such as Value-at-Risk (VaR) andExpected Shortfall (ES) cannot capture the risk exposurecaused by the co-movement of multiple financial variables.

Use quantitative definition of the exposure on the tail regionsto model the dependence of some financial variables




No formal definition/principle about what is the standard ofrisk measure when considering the co-movement infinancial/insurance industry ⇒ hard to extend

How to consider the choice under the extreme risk whichquantitatively evaluates the preference of tail risk ? ⇒ Useutility theory and theoretical aspects of distortion functionswhen facing high risk events

Asymptotic theory of the risk measures on tail regions




Our Contribution

Compare risks under some extreme scenarios with differentrisk levels.

Consider the risk exposure on the tail regions where abenchmark loss is included in the modeling of extremescenarios.

Explore a limiting space of the above risks on tail regions asthe extreme scenarios go to extremes.

Explore distortion risk measure on tail regions and itsasymptotic property. Establish asymptotic results usingExtreme Value Theory and tail copulas.

Nonparametric estimator has been proposed and itsasymptotic normality is proved.




Tail Risks under the Extreme ScenariosExtreme Preference and Its Distortion

Preference Relationship Components

X : the risk random variable

FX : survival distribution

Γ: a given collection of survival distribution FX

V: the set of all non-negative risk random variables, definedon space Γ. V always represents the loss.

(X ,Y ): a pair of random losses with continuous jointdistribution function F (x , y) and marginal distributionsF1(x) = F (x ,∞) and F2(y) = F (∞, y).

Qi = F−1i , i = 1, 2: the inverse functions of marginal

distributions.

α: the level of the risk, α ∈ [0, 1]





The preference relationship

The relations, indifference ∼ and preference �, for any two(survival) distributions F , G in Γ are

F ∼ G if and only if F (t) = G (t), ∀ t ≥ 0.

F � G if and only if F (t) ≥ G (t), ∀ t ≥ 0. (1)

The preference � on V is based on distortion function g is

X1 � X2 if and only if

∫ ∞−∞

g(FX1(t))dt ≥∫ ∞−∞

g(FX2(t))dt,

The indifference ∼ for X1,X2 in V is

X1 ∼ X2 if and only if

∫ ∞−∞

g(FX1(t))dt =

∫ ∞−∞

g(FX2(t))dt.





Preference relations on the tail risks

Suppose Y is a benchmark whose extreme scenarios are ofinterest.

Consider the univariate case where extreme scenarios of Y areincluded as conditional events, and the associated preferencerelation is called extreme preference when the extremescenarios go to extreme.

Technique: Extreme preference depends on the transformationof three spaces in the following sections (from Γ× C, to Γ′,then to Γ0).





Two extensions:

First, define the scenarios of Y as an event S(Y , α) where αrepresents the level of the risk.Typically, choose S(Y , α) = {Y > Q2(1− α)} as the extremescenarios.

Second, truncate the non-tail risk of X below the thresholdQ1(1− α) and consider the ratio of this truncated variableand the threshold.





Space Construction

Extreme scenarios event S(Y , α) = {Y > Q2(1− α)}Truncate the non-tail risk of X below the threshold Q1(1− α)and consider the ratio of this truncated variable and thethreshold.

Construct space V is the collection, or a subcollection, of Xhaving distribution F with extreme index γ > 0, whichsatisfies

limt→∞

F (tx)

F (t)= x−1/γ , x > 0. (2)

Γ: the collection of all survival distributions of X





C: a collection of (survival) copulas of (X ,Y ), we may say“given benchmark Y ” and “given the survival copula of(X ,Y )” equivalently in the following as the scenarios of Y is{Y > Q2(1− α)}.Consider the conditional survival distribution

P(

(X−Q1(1−α))+

Q1(1−α) > t∣∣∣ Y > Q2(1− α)

)=

{C(αsα((1+t)−1/γ),α)

α t ≥ 0

1 t < 0

where a+ = max(a, 0) for any real value a, b and

sα(x) = F (Q1(1−α)x−γ)α such that sα(0) = 0, sα(1) = 1 and

sα(∞) = 1/α.





Define space Γα and its elements Gα (Fix α)

Get Gα by a pair F and C such that

Gα(x) =

{C(αsα(x−1/γ),α)

α x ≥ 1

1 0 ≤ x < 1(3)

Gα ∈ Γα is induced by a pair of F and C .





combine Γ′ = ∪αΓα and V′ = ∪αVα and consider thepreference relations on them.

for any G1α1 ,G2α2

G1α1 ∼ G2α2 if and only if G1α1(t) = G2α2(t), ∀ t ≥ 0.

G1α1 � G2α2 if and only if G1α1(t) ≥ G2α2(t), ∀ t ≥ 0.(4)





Axiom 1 (Neutrality) Let Y be a given loss, X1 and X2 belong to Vwith respective levels α1, α2 and functions G1α1 and G2α2 inΓ′ . Then,

G1α1 ∼ G2α2 =⇒ X1|(Y , α1) ∼ X2|(Y , α2) on V′

where X |(Y , α) means the loss X given the dependence withY and level α.

Axiom 2 (Complete weak order) � on Γ′ is reflexive, transitive, andconnected.

Axiom 3 (Continuity) � on Γ′ is continuous with respect to L1-norm.Axiom 4 (Monotonicity) If for any t ≥ 1, G1(t) ≥ G2(t), then G1 � G2

on Γ′.Axiom 5 (Dual Independence) If G ,G ′,H belong to Γ′ and ω is any

real value satisfying ω ∈ [0, 1], then G � G ′ impliesωG ⊕ (1− ω)H � ωG ′ ⊕ (1− ω)H where ⊕ is the harmonicconvex combination operator.





Theorem 1

Theorem

For any fixed level α1, α2 ∈ (0, 1), a preference relation � satisfiesAxioms A1-A5 if and only if there exists a distortion function gsuch that given the loss Y (given any survival copulasrespectively), for any X1,X2 in V with respective G1α1 ,G2α2 in Γ′,

X1|(Y , α1) � X2|(Y , α2) on V′ if and only if∫ ∞0

g(G1α1(t))dt ≥∫ ∞

0g(G2α2(t))dt.

Moreover, any variable X in V is equivalent to its utility on V′given Y , α and thus Gα ∈ Γ′ so that

X |(Y , α) ∼ [Uα(X ;Y ); 1] |(Y , α) on V′. (5)

where Uα(X ;Y ) :=∫∞

0 g(Gα(t))dt and [x ; p] is a binary randomvariable taking x with probability p and 0 with probability 1− p.





Asymptotic limit of the extreme preference as α ↓ 0Three questions:

1) Is there a collection of distributions which represent the limitsof Gα in Γ′ as α ↓ 0?

2) If there exists such a collection Γ0 for which similar Axioms1-5 hold as well?

3) If there exists such a collection Γ0 satisfying some axiomssimilar to Axioms 1-5, what is the distortion function? Is itconsistent with the g in Theorem 1?





Axiom 0’ Suppose there exists a measure R : (0,∞)2 → [0,∞) suchthat

limα↓0

C (αsα(x), α)

α= R(x , 1), x ≥ 1.

Axiom 1’ (Neutrality) Let Y be a given loss, X1 and X2 belong to Vwith respective G1 and G2 in Γ0 . Then,

G1 ∼ G2 =⇒ X1|(Y ) ∼ X2|(Y ) on V0.

Axiom 2’ (Complete weak order) � on Γ) is reflexive, transitive, andconnected.

Axiom 3’ (Continuity) � on Γ0 is continuous with respect to L1-norm.Axiom 4’ (Monotonicity) If for any t ≥ 1, G1(t) ≥ G2(t), then G1 � G2

on Γ0.Axiom 5’ (Dual Independence) If G ,G ′,H belong to Γ0 and ω is a real

value satisfying ω ∈ [0, 1], then G � G ′ impliesωG ⊕ (1− ω)H � ωG ′ ⊕ (1− ω)H where ⊕ is the harmonicconvex combination operator.





Theorem 2

Theorem

A preference relation � satisfies Axioms A0’-A5’ if and only ifthere exists a distortion function g0 such that given the loss Y(given any survival copulas respectively), for any X1,X2 in V withrespective G1,G2 in Γ0,

X1|(Y ) � X2|(Y ) on V0 if and only if∫ ∞0

g0(G1(t))dt ≥∫ ∞

0g0(G2(t))dt.

Moreover, any variable X in V is equivalent to its utility on V0

given Y and thus G ∈ Γ0 so that

X |(Y ) ∼ [U0(X ;Y ); 1] |(Y ) on V0. (6)

where U0(X ;Y ) :=∫∞

0 g0(G (t))dt and [x ; p] is a binary randomvariable taking x with probability p and 0 with probability 1− p.




Distortion Risk Measure DifinitionStatistical Inference

Definition

Let ρg : L(R+)→ R+ be a distortion risk measure for a univariatenon-negative random loss Z with distribution function FZ ∈ L(R+)and non-decreasing distortion function g satisfyingg(0) = 0, g(1) = 1, the corresponding distortion risk measure is:

ρg (Z ) =

∫ ∞0

g(1− FZ (x))dx .





Definition

The distortion risk measure on tail regions of (X ,Y ) is thedistortion risk measure ρg applied to the distribution of(X−Q1(1−α))+

Q1(1−α) |Y > Q2(1− α) taking into account the marginalthreshold, which is

ρg (X ;Y , α) = Q1(1−α)+

∫ ∞Q1(1−α)

g (P(X > x |Y > Q2(1− α))) dx

where g is a (right-continuous) distortion function.





Nonparametric Estimation

(X1,Y1), . . . , (Xn,Yn):independent and identically distributedrandom vectors with joint distribution F .X1:n ≥ X2:n ≥ . . . ≥ Xn:n,Y1:n ≥ Y2:n ≥ . . . ≥ Yn:n.Xt:n = Xbtc:n where b·c is the flooring integer of any positivereal values.

F1(x) = F (x ,∞), F2(y) = F (∞, y) and Qi = F−1i , i = 1, 2,

the generalized inverse function of Fi .

Fn1(x) = 1n

∑ni=1 I (Xi > x), Fn2(y) = 1

n

∑ni=1 I (Yi >

y), x , y ∈ R.: the empirical survival functions





The smoothed and non-smoothed empirical (survival) copulas aregiven by{

C (u, v) = 1n

∑ni=1 I (Fn1(Xi ) < u, Fn2(Yi ) < v),

C (u, v) = 1n

∑ni=1 K

(1−Fn1(Xi )/u

h

)K(

1−Fn2(Yi )/vh

), u, v ∈ [0, 1].

where h := hn > 0 is the bandwidth.Two nonparametric estimators of ρg (X ;Y , α) defined below

ρg (X ;Y , α) = 1nα

∑ni=1(Xi − Xnα:n)I (Fn1(Xi ) < α)

(g(C(Fn1(Xi ),α)

α

)− g

(C(Fn1(Xi )− 1

n,α)

α

))+Xnα:n

ρg (X ;Y , α) = 1nα

∑ni=1(Xi − Xnα:n)I (Fn1(Xi ) < α)

(g(C(Fn1(Xi ),α)

α

)− g

(C(Fn1(Xi )− 1

n,α)

α

))+Xnα:n




Assumptions

(1.a) There exist some γ1 ∈ (0, 1), β1 ≤ 0 and function A which isslowly regular varying with index β1 and a constant sign nearinfinity, such that

limt→∞

1

A(1/F1(t))

(F1(tx)

F1(t)− x−1/γ1

)= x−1/γ1

xβ1/γ1 − 1

γ1β1, x > 0.

(1.b) There exist functions R : (0,∞)2 → [0,∞) and0 < δ0 < 1, L > 0, τ < 0, β > γ1,T > 1 such that, as t →∞

supx∈(0,∞),y∈(0,T ]

(tC (t−1x , t−1y)− R(x , y)

xβ

)= O(tτ ).

and

supx ,∈(0,δ0],y∈(0,1]

R(x , y)

(x ∧ y)β≤ L.

(1.c) There exists 0 < δ′0 < 1, L′ > 0 such that

|g(u)− g(u′)| ≤ L′|u − u′|, ∀u, u′ ∈ (0, δ′0).

(1.d) As n→∞, α = O(n−1+κ) for some 0 < κ < −2τ−2τ+1 and√

nαA(α−1) = o(1).




Theorem

Suppose the set B = {x ∈ (0, 1] | g is discontinuous at R(x , 1)}has zero Lebesque measure. Under the Assumption (1.a)-(1.c), thedistortion risk measure satisfies

limα↓0

ρg (X ;Y , α)

Q1(1− α)= 1 +

∫ ∞1

g(R(x−1/γ1 , 1))dx . (7)

Theorem

Suppose the set B = {x ∈ (0, 1]) | g is discontinuous at R(x , 1)}has zero Lebesque measure. Under the Assumption (1.a)-(1.d), thedistortion risk measure satisfies

limn→∞

√nα

∣∣∣∣ρg (X ;Y , α)

Q1(1− α)− ρ0

∣∣∣∣ = 0. (8)




Assumptions:

(2.a) The function R in Assumption (1.b) has a continuousfirst-order partial derivatives R1(x , 1) = ∂R(x , 1)/∂x .

(2.b) R(1, 1) > 0 and R1(1, 1) > 0.

(2.c) The kernel distribution function K has symmetric density kwith support [−1, 1] and the kernel density k has boundedfirst derivative.

(2.d) The bandwidth h := hn > 0 satisfies nαh2 →∞ andnαh4 → 0.




Theorem

Let α = αn be an intermediate sequence such that αn → 0 andnαn →∞ as n→∞. Suppose the setB = {x ∈ (0, 1] | g is discontinuous at R(x , 1)} has zero Lebesquemeasure. Under the Assumptions ?? and (2.a), (2.b),

√nα

(ρg (X ;Y , α)

ρg (X ;Y , α)− 1

)d−→ N(0, σ2

R,g ,γ1).

with σ2R,g ,γ1

= Var(ΦR,g ,γ1)/ρ20 and

ΦR,g ,γ1 = γ1

∫ 1

0

WR(R←(x), 1)

(R←(x))γ1+1R1(R←(x), 1)

dg(x − γ1WR(R←(1), 1)

(R←(1))γ1+1R1(R←(1), 1).




Theorem continued

Theorem

is non-degenerate. R← is the (right-continuous) generalizedinverse function of R(·, 1) and WR is a R-Brownian motion, i.e. azero-mean Gaussian process with covariance function with

E(WR(u1, v1)WR(u2, v2)) = R(u1 ∧ u2, v1 ∧ v2),

(ui , vi ) ∈ (0,∞]2\{∞,∞}, i = 1, 2.

Moreover, if Assumption (2.c) and (2.d) are also satisfied, then

√nα

(ρg (X ;Y , α)− ρg (X ;Y , α)

ρg (X ;Y , α)

)P−→ 0.




Q & A ?




Thank you very much for your attention!


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Extreme Preference and Distortion Risk Measures on Tail ......the extreme scenarios go to extremes....

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